Does solar physics provide constraints to weakly interacting massive particles?
A. Bottino,1,*G. Fiorentini,2,†N. Fornengo,1,‡B. Ricci,2,§S. Scopel,1,储 and F. L. Villante2,¶ 1Dipartimento di Fisica Teorica, Universita` di Torino Istituto Nazionale di Fisica Nucleare, Sezione di Torino via P. Giuria 1, I–10125 Torino, Italy
2Dipartimento di Fisica, Universita` di Ferrara, Istituto Nazionale di Fisica Nucleare, Sezione di Ferrara via del Paradiso 12, I–44100 Ferrara, Italy
共Received 26 June 2002; published 23 September 2002兲
We investigate whether present data on helioseismology and solar neutrino fluxes may constrain weakly interacting massive particle共WIMP兲-matter interactions in the range of WIMP parameters under current ex-ploration in WIMP searches. We find that, for a WIMP mass of 30 GeV, once the effect of the presence of WIMPs in the Sun’s interior is maximized, the squared isothermal sound speed is modified, with respect to the standard solar model, by at most 0.4% at the Sun’s center. The maximal effect on the 8B solar neutrino flux is
a reduction of 4.5%. Larger masses lead to smaller effects. These results imply that present sensitivities in the measurements of solar properties, though greatly improved in recent years, do not provide information or constraints on WIMP properties of relevance for dark matter. Furthermore, we show that, when current bounds from direct WIMP searches are taken into account, the effect induced by WIMPs with dominant coherent interactions is drastically reduced as compared to the values quoted above. The case of neutralinos in the minimal supersymmetric standard model is also discussed.
DOI: 10.1103/PhysRevD.66.053005 PACS number共s兲: 95.35.⫹d, 98.35.Gi, 98.62.Gq
I. INTRODUCTION
A host of independent astronomical observations point to the existence in our Universe of a total amount of matter in the range 0.2ⱗ⍀mⱗ0.4 关1兴 共or equivalently, 0.05ⱗ⍀mh2
ⱗ0.3, where h is the Hubble constant in units of
100 km s⫺1 Mpc⫺1), well beyond the amount of visible matter ⍀vis⬃0.003. Since the primordial nucleosynthesis tells us that baryons cannot contribute for more than about 5% 关2兴, most of the dark matter must be nonbaryonic. The evolution theory of the primordial density fluctuations into the present cosmological structures indicates that most of the dark matter must be comprised of cold particles, i.e., of par-ticles that decoupled from the primordial plasma when non-relativistic. A particle with the suitable properties for being a significant cold dark matter relic is generically defined as a WIMP共weakly interacting massive particle兲.
A variety of physical realizations for a WIMP are offered by various extensions of the standard model 关3兴, the neu-tralino being one of the most appealing candidates. In the present paper, many considerations are developed in terms of generic WIMPs. We consider the neutralino, whenever we wish to narrow down to a specific candidate.
WIMPs are very actively searched for by means of vari-ous experimental strategies. Direct searches rely on the mea-surements of the signal that a nucleus of an appropriate de-tector would generate, when hit by a WIMP 关4兴. Indirect
searches are based on measurements of the signals due to WIMP pair annihilations in the galactic halo or inside celes-tial bodies 关3兴.
The event rates of WIMP direct search experiments are proportional to the product of the local WIMP density in the Galaxy times the WIMP-nucleus cross section. In what follows will be expressed as ⫽•l, where l is the
local total dark matter density and (⭐1) is a scaling parameter which accounts for the actual fraction of local dark matter to be ascribed to the candidate.
WIMP-matter interactions are conveniently classified in terms of coherent cross sections and 共nuclear兲 spin-dependent ones. In the first case, the WIMP-nucleus cross section is given in terms of the proton and WIMP-neutron cross sections. To simplify the formalism, in the fol-lowing we further assume that, in the coherent case, WIMPs interact with equal strength with protons and neutrons 共for instance, this may safely be assumed for neutralinos, while it is not the case for neutrinos兲, and then we express a generic WIMP-nucleus coherent cross section in terms of a single WIMP-nucleon cross section c. For the spin-dependent
case, the derivation of a WIMP-nucleon cross section from the WIMP-nuclear one depends on the nature of the WIMP and on specific nuclear properties 关5兴.
In the case of WIMPs with coherent interactions with matter, the range of WIMP parameters under current explo-ration in WIMP direct searches is conveniently expressed in terms of the quantityc:
10⫺43 cm2⭐c⭐6⫻10⫺41 cm2, 共1兲 for WIMP masses m in the range
30 GeV⭐m⭐270 GeV. 共2兲
In the derivation of the ranges in Eqs. 共1兲, 共2兲, one has taken into account a variety of WIMP distribution functions *Electronic address: [email protected]
†Electronic address: [email protected] ‡Electronic address: [email protected];
URL: http://www.to.infn.it/⬃fornengo
§Electronic address: [email protected]
储Electronic address: [email protected]
in the galactic halo and uncertainties in the determination of the relevant astrophysical quantities关6,7兴.
For the reasons mentioned above, in the case of spin-dependent interactions no model-inspin-dependent sensitivity range may be derived. However, to give an indication, we recall that, in the case of a relic particle which interacts with matter mainly by spin-dependent interactions mediated by Z-boson exchange, current WIMP direct experiments are sensitive to values of sd of about 10⫺36⫺10⫺37cm2 关5兴,
wheresd is the spin-dependent WIMP-proton cross section.
One of the WIMP direct search experiments, the DAMA/ NaI共Tl兲 experiment 关8兴, has observed an annual modulation effect 共at a 4 C.L.兲 with all the features expected for a WIMP signal 关9兴. When interpreted as due to a WIMP with coherent interactions, the DAMA effect provides a 3region in the plane m-cembedded in the range of Eqs.共1兲, 共2兲.
In Ref.关10兴 it was proved that this annual modulation effect is compatible with an interpretation in terms of relic neutrali-nos共see also Ref. 关11兴兲.
If WIMPs populate our Galaxy, they can be captured by the Sun and accumulate in its core. For suitable values of the WIMP parameters, this could have relevant effects on the Sun. In fact, WIMPs could provide an effective mechanism for energy transport in the Sun, producing an isothermal core and reducing substantially the Sun central temperature Tc.
Following this idea, some time ago a special class of WIMPs named cosmions, with masses of a few GeV and scattering cross sections on nucleons of the order of 10⫺36-10⫺34cm2, was studied in detail as a way of solving simultaneously the solar neutrino puzzle and the dark matter problem 共see e.g., Refs. 关12–17兴兲. The cosmion hypothesis was progressively abandoned when it became clear that the solar neutrino puzzle cannot be accounted for by simply reducing Tc.
In the last twenty years, our observational knowledge of the solar interior has progressed enormously. By means of helioseismic data it has become possible to derive the sound speed with an accuracy of about one part per thousand over most of the radial profile and of about one percent in the innermost part 关18兴. By the same method, it has been pos-sible to deduce important properties of the convective enve-lope. The photospheric helium fraction Yph and the depth of
the convective envelope Rb have been determined with an
accuracy of about one percent and one per thousand respec-tively, following the pioneering papers of Refs. 关19,20兴. Moreover, the measurement of the neutrino flux from 8B decay obtained by combining关21–24兴 SNO charged current
关25兴 and super-Kamiokande 关26兴 data and confirmed by the
recent SNO neutral current results 关27兴, has provided a de-termination of the temperature Tcnear the center of the Sun
at the level of about one percent 关28兴.
All the predictions of the standard solar model 共SSM兲 have been confirmed by these accurate tests, so one is natu-rally led to the question of whether our accurate knowledge of the Sun interior can be used to constrain the WIMP pa-rameter space. This particularly interesting question was re-cently raised in Refs.关29兴 and 关30兴, where it was concluded that solar physics can be used to significantly constrain the WIMP parameter space.
In this paper, we provide an alternative analysis of the problem, obtaining results substantially different from those derived in Refs.关29,30兴.
The plan of the paper is as follows. In Sec. II we discuss WIMP energy transport in the Sun, introducing the relevant physical parameters to be discussed in the subsequent sec-tions. In Secs. III and IV WIMP energy transport is applied to the central solar structure, in order to determine the re-gions of the WIMP parameter space where the WIMP energy transport is most efficient. This leads to an optimal choice of WIMP parameters which is then used in Sec. V to calculate the solar sound speed profile and discuss the possible impact of helioseismology on WIMP properties. The sensitivity of the available data on 8B solar neutrinos to the presence of WIMPs in the Sun is then discussed in Sec. VI. The neu-tralino, as a specific realization of WIMP, is discussed in Sec. VII. Section VIII is devoted to our conclusions.
II. WIMP ENERGY TRANSPORT
In this section, we briefly review some issues relevant to the problem of WIMP energy transport in the Sun addressed in Refs.关12–17兴.
WIMPs are captured by the Sun during the course of its lifetime and are confined by the effect of gravity in a central region with a radial number density distribution approxi-mately given by 关31,32兴 共we hereby use natural units: ប⫽c
⫽kB⫽1)
n共r兲⫽共n兲cexp
冉
⫺r 2r2
冊
, 共3兲 where the quantity (n)c⫽(1/3/2)(N/r3) is the central WIMP number density, Nis the total number of WIMPs in the Sun andr⫽
冉
3Tc 2Gcm冊
1/2
, 共4兲
Tcandcbeing the Sun central temperature and density, and m the WIMP mass. For Tc⫽1.57⫻107 K and c ⫽154 g cm⫺3, one finds r
⯝0.01(100 GeV/m)1/2R䉺. We note that Eq.共3兲 is obtained under the assumption of constant density and temperature in the region of the Sun populated by WIMPs. This approximation is accurate enough for our purposes.
WIMPs provide a mechanism for energy transport in the solar core whose efficiency depends on the Knudsen param-eter
K⫽l共0兲
r , 共5兲
where l(0) is the WIMP mean free path near the solar cen-ter and generally
1
l共r兲⫽
兺
i iXi共r兲共r兲 mi
Here the sum extends to all the nuclear species present in the Sun, Xi is the mass fraction of the ith element,(r) is the
density profile of the Sun and i is the WIMP scattering
cross section with the nucleus of species i.
For KⰆ1 the WIMP mean free path is much smaller than the dimension of the region where the WIMPs are confined and the energy transfer mechanism is conductive. The prob-lem of thermal conduction by a dilute gas of massive par-ticles is discussed in Ref.关12兴. For our purposes, it is useful to express the WIMP luminosity in the conductive regime by using the radiative transport equation:
L,cond共r兲⫽⫺16arad 3 T共r兲3r2 共r兲共r兲 dT dr, 共7兲 where the WIMP opacityis defined as
1 共r兲⫽ 3n共r兲 4aradT共r兲3
冑
T共r兲 m共r兲l共r兲. 共8兲 In Eqs.共7兲, 共8兲 the quantity araddenotes the radiation density constant andis a dimensionless coefficient which depends on the ratio between the WIMP mass and the masses of back-ground nuclei.1For KⰇ1 the WIMP mean free path is much larger than the region in which WIMPs are trapped and the energy trans-fer mechanism is nonlocal. An analytic approximation to treat this case has been developed by Spergel and Press关13兴 who described WIMPs as an isothermal gas at an appropriate temperature T. In this case, the luminosity L,NLcarried by WIMPs can be expressed as
L,NL共r兲⫽
冕
0r
dr
⬘
4r⬘
2共r⬘
兲⑀SP共r
⬘
;T兲, 共9兲 where ⑀SP represents the energy transferred to WIMPs per second and per gram of nuclear matter and is given by⑀SP共r;T兲⫽8
冑
2 n共r兲兺
i i Xi共r兲 mi mmi 共m⫹mi兲2 ⫻冉
mT共r兲⫹miT mmi冊
1/2 关T共r兲⫺T兴. 共10兲 The WIMP temperature T is fixed by requiring that the energy absorbed by WIMPs in the inner region 关where T(r)⬎T兴 is equal to that released in the outer region 关where T(r)⬍T兴: i.e.,L,NL共R䉺兲⫽
冕
0R䉺
dr
⬘
4r⬘
2共r⬘
兲⑀SP共r⬘
;T兲⫽0. 共11兲Gould and Raffelt 关14兴, by means of a Monte Carlo inte-gration of the Boltzmann collision equation, showed that the Spergel and Press approximation, Eq.共10兲, overestimates the energy transfer typically by a factor of a few. Moreover, by studying simplified stellar models, they showed that it is pos-sible to approximate energy transfer in the nonlocal regime by a conductive treatment if one applies a global ‘‘luminosity suppression factor.’’ We follow their approach by using as a general expression for the WIMP luminosity:
L共r兲⫽ f 共K兲L,cond共r兲, 共12兲
where
f共K兲⫽ 1
共K/K0兲2⫹1
, 共13兲
and K0⫽0.4 is the value of the Knudsen parameter for which the WIMP energy transport is most efficient. We remark that other approaches are possible. In Refs. 关15兴 and 关29,30兴, for example, a suitable interpolation of Eqs.共7兲 and 共9兲 is used as a general expression for WIMP luminosity. We have checked numerically that our results are essentially indepen-dent of the chosen approach.
As mentioned before, WIMPs can interact with nuclei ei-ther via coherent interactions or via spin-dependent ones. In the first case the WIMP cross section on the ith nucleus of mass miis related to the corresponding WIMP-nucleon cross
section, c, by i⫽c
冉
mi mn冊
4冉
m⫹mn m⫹mi冊
2 , 共14兲where mn is the nucleon mass. With this assumption, Eq.共6兲 may be rewritten as 1 l共r兲⫽ 1 ln共r兲
兺
i Xi共r兲冉
mi mn冊
3冉
m⫹mn m⫹mi冊
2 , 共15兲 with ln共r兲⫽ mn c共r兲 ⯝冉
1.6⫻10⫺37 cm 2 c冊冉
150 g cm⫺3 共r兲冊
•R䉺. 共16兲In the Sun the abundances Xi for nuclei heavier than
hydro-gen and helium are much smaller than those of H and He. However, in the sum over elements of Eq. 共15兲, for heavy nuclei 共mainly for iron兲 the reduction due to Xi is largely compensated by the coherence effect displayed by Eq. 共14兲. On the other hand, as far as spin-dependent interactions are concerned, the contribution of hydrogen is overwhelm-ingly dominant over the contributions of other nuclear spe-cies with nonvanishing spins. Therefore, from now on we identify sd with the spin-dependent WIMP-proton cross
section.
In the following all WIMP matter interactions will be ex-pressed in terms of c and/orsd. The notationp will be
used to denote a generic WIMP-proton cross section;p has
1The coefficient
can be derived from the values of the thermal conduction coefficient (m/mi) tabulated in Ref. 关12兴. It holds:
⫽(1/l)关兺i(1/l,i)兴⫺1 where l,iis the WIMP mean free path relative to the nucleus i.
to be identified withc共orsd), whenever coherent共or spin
dependent兲 interactions are considered.
III. WIMPS AND THE CENTRAL SOLAR STRUCTURE The efficiency of WIMP energy transport depends on sev-eral parameters. Specifically, it depends on the total number Nof WIMPs in the Sun, on the WIMP mass mand on the WIMP-nucleon cross section p. In order to determine the
region of the WIMP parameter space where the WIMP en-ergy transport is most efficient it is useful to define the quan-tity
␦⫽ lim r→0
L共r兲
L共r兲 , 共17兲
where L is the radiative luminosity of the Sun. It is easy to show that
␦⫽
冋
1共K/K0兲2⫹1
册
␥共0兲
共0兲, 共18兲
where␥,(0) are the opacities at the solar center for radia-tion and WIMPs.
In Figs. 1 and 2 we show the lines in the plane (p,N)
which correspond to fixed values of ␦. We have chosen m
⫽50 GeV as a representative value for the WIMP mass.
Fig-ure 1 refers to coherent WIMP scattering, while Fig. 2 is obtained for spin-dependent WIMP interactions. The lines have been calculated by using the physical and chemical pa-rameters of the Sun as they are obtained by a standard evo-lutionary code 关33兴.
The qualitative behavior of the iso-␦ lines is easily under-stood. The efficiency of the WIMP energy transport is pro-portional to 1/p共i.e., to K) in the conductive regime and to p 共i.e., to K⫺1) in the nonlocal regime. The transition
be-tween the two regimes occurs at K⫽l/r⯝0.4, i.e., at a cross sectiont which corresponds, for coherent scattering, to the valuet⯝10⫺37cm2 and, for spin-dependent interac-tions, to t⯝10⫺34cm2. The difference in the values of the transition cross sections t is due to the fact that, for the
spin-independent case, interactions on elements heavier than hydrogen are important, due to the coherence effect in the scattering which is manifest in Eq. 共14兲, while for spin-dependent interactions no coherence is present and therefore only scattering on hydrogen共which is largely more abundant than other elements with nonvanishing spin兲 matters.
To be more quantitative, we can also show the analytic expression of the iso-␦ lines as a function ofp:
N共␦,p兲⫽N共␦兲关共p/t兲⫹共p/t兲⫺1兴, 共19兲
where N(␦) is given by the condition
␥共0兲⫽␦共0;l⫽0.4r兲, 共20兲 which gives N共␦兲 3/2r 3⫽␦ 4aradTc 3 3
冑
m Tc l␥ 0.4r, 共21兲 and l␥⯝5⫻10⫺3 cm is the photon mean free path at the center of the Sun.We remark that, since WIMP radial distribution is rapidly decreasing关see Eq. 共3兲兴, the parameter␦ essentially provides an upper limit for the ratio L/L in the Sun. As a conse-quence, the condition that ␦ is not too small should be used as a necessary (but not sufficient) condition for WIMPs to have relevant effects on the Sun. This means the following.
共i兲 We can safely conclude that regions of the parameter
space for which ␦ⱗ10⫺2are not interesting. This point can be easily understood. WIMPs modify the relation between the luminosity and the gradient of temperature in the Sun. This essentially corresponds to a redefinition of the Sun ra-diative opacity␥. The effect is maximal at the center of the Sun where one has␥→␥/(1⫹␦). The radiative opacity is affected by uncertainties at the level of a few percent 关34兴. This means that we cannot disentangle WIMP effects if␦ is smaller than 10⫺2.
共ii兲 We cannot conclude, at this stage, that regions in
which ␦ is large (say e.g. ␦⬎1) correspond to sizeable
FIG. 1. The solid curves denote the total number Nof WIMPs accumulated in the Sun’s core due to gravitational capture as a function of the WIMP-nucleon cross section, for a WIMP mass of 50 GeV and for the case of purely coherent WIMP-nucleus interac-tions. The thick solid line refers to WIMPs which do not annihilate once captured. The thin solid lines refer to annihilating WIMPs, for increasing values of the zero-temperature thermally averaged WIMP-antiWIMP annihilation cross section: 具av典0⫽10⫺30,
10⫺27, 10⫺24, 10⫺21cm3s ⫺1. The dot-dashed lines show the con-tours of constant values of the parameter ␦, defined in Eq. 共17兲, which quantifies the efficiency of energy transport of WIMPs with respect to photons at the center of the Sun. The vertical dashed line denotes the upper limit on the WIMP-nucleon elastic cross section for coherent interactions. The large dot identifies the WIMP param-eters for which the WIMP effect on the Sun’s energetic is maximal.
modifications of the solar structure. The parameter ␦ is, in fact, a local parameter. A large ␦ only implies that WIMPs give a large contribution to the energy transport at the center of the Sun. In order to understand whether WIMPs have relevant effects on global properties of the Sun, we also have to compare the radius r of the WIMPs extension region with the relevant length scales in the Sun共e.g., the dimension of the neutrino production region, the temperature height scale兲.
IV. THE NUMBER OF WIMPS IN THE SUN Up to this point we have treated m,p and N as
inde-pendent quantities. However, this is not the case, as noted long ago in Ref. 关16兴. N depends on the capture rate of the halo WIMPs by the Sun during its lifetime; in turn, this capture rate is proportional to the product of the WIMP den-sity in the halotimesp, i.e., to the productp.
More-over, captured WIMPs may annihilate with captured anti-WIMPs. Therefore Nalso depends on the quantity
具
av典
0, where a is the WIMP-antiWIMP pair annihilation crosssection andv is the relative velocity of the annihilating pair,
while brackets indicate the average over the WIMP velocity distribution in the Sun core. Explicitly, one has关31兴
N⫽⌫ctanh共t䉺/兲, 共22兲 where t䉺⯝4.5 Gyr is the age of the Sun, ⌫c is the WIMP capture rate and⫽1/
冑
⌫cCais a time scale parameter which determines whether and when equilibrium between capture and annihilation occurs. The quantity Ca is given byCa⫽
具
av典
0 V0冉
m 20 GeV冊
3/2 , 共23兲with V0⯝2.3⫻1028cm3 for the Sun. In the limit in which
具
av典
0→0, one has→⬁ and Eq. 共22兲 becomesN⫽⌫ct䉺⬅Nmax. 共24兲 Nmaxrepresents the upper limit for the WIMP number in the Sun at the present time.
The expression of the capture rate ⌫c is rather involved. For a Maxwellian WIMP velocity distribution in the galactic halo it may be cast in the form 关35兴
⌫c⫽
兺
i冉
8 3冊
1/2冋
i mv¯册冋
Mi mi册
冋
3vesc 2 2¯v2具
典
i册
共⬁兲Si, 共25兲where the cross sectionsi are related top as discussed in
the previous section, Mi is the total mass of the ith nucleus
in the Sun,vesc⯝618 km sec⫺1is the escape velocity at the Sun’s surface and ¯v⫽270 km sec⫺1 is the WIMP velocity dispersion. The quantities
具
典
i denote the reducedgravita-tional potential (r)⫽vesc2 (r)/vesc2 averaged over the mass distribution of the ith nucleus, as defined in Eq.共A2兲 in the Appendix. The quantity (⬁)⯝0.75 is a suppression factor due to the motion of the Sun through the halo and the factor Siincludes the effects of WIMP-nucleus mass mismatch and
finite target dimensions共note that thei are point-like cross sections兲. The expression of Si is given in the Appendix for completeness. The chemical composition of the Sun and the values of the quantities
具
典
i, which have been used in ourcalculations, are given in Table I. We note that for relatively heavy WIMPs like the ones which are considered in this
FIG. 2. The same as in Fig. 1, for spin-dependent WIMP-nucleus interactions. The vertical dotted line denotes the upper limit on the WIMP-nucleon cross section for Z-boson mediated spin-dependent interactions.
TABLE I. Sun composition, average mass fractions of the chemical elements in the Sun as obtained from Ref.关44兴 and values of the average reduced gravitational potential具典i.
Element Xi 具典i H 0.71 3.16 He 0.27 3.40 C 0.30⫻10⫺2 3.23 N 0.93⫻10⫺3 3.23 O 0.84⫻10⫺2 3.23 Ne 0.17⫻10⫺2 3.23 Na 0.35⫻10⫺4 3.23 Mg 0.65⫻10⫺3 3.23 Al 0.56⫻10⫺4 3.23 Si 0.70⫻10⫺3 3.23 P 0.62⫻10⫺5 3.23 S 0.37⫻10⫺3 3.23 Cl 0.78⫻10⫺5 3.23 Ar 0.94⫻10⫺4 3.23 Ca 0.65⫻10⫺4 3.23 Ti 0.36⫻10⫺5 3.23 Cr 0.17⫻10⫺4 3.23 Mn 0.96⫻10⫺5 3.23 Fe 0.12⫻10⫺2 3.23 Ni 0.74⫻10⫺4 3.23
analysis (mⲏ30 GeV) the capture in the Sun for spin-independent interaction is not dominated by scattering on hydrogen, even though this is the most abundant element, but instead by interactions on heavier elements. For instance, for a 50 GeV WIMP, capture is dominated by He, O and Fe. The smaller abundance of heavier elements is largely compen-sated by the coherence effect in the cross section shown in Eq. 共14兲. On the contrary, in the case of spdependent in-teractions the only relevant capture process is on hydrogen, since in this case no coherence effect is present and therefore capture is dominated by interactions with the most abundant element which possesses spin, i.e., hydrogen.
Equation 共25兲 holds when the cross section p is small
enough so that multiple scatterings can be neglected. How-ever, whenp is so large that every WIMP crossing the Sun is captured,⌫csaturates to a maximal value, and Eq.共25兲 is replaced by ⌫c,sat⫽
冉
38冊
1/2冋
m¯v册冋
Mi mi册
冋
⫹ 3vesc 2 2¯v2册
共⬁兲R䉺 2 , 共26兲where⯝1.77. In Eq. 共26兲, the first term in the last square brackets refers to WIMPs whose orbit crosses the Sun even without gravitational deflection, while the second term is a gravitational focusing factor and corresponds to those WIMPs whose orbits pass through the Sun because of gravi-tational deflection. One can easily see that for the Sun the focusing factor is dominant over the purely ‘‘geometrical’’ one, since the velocity dispersion of WIMPs in the Galaxy is a factor of 2 smaller than the escape velocity at the Sun’s surface.
The behavior of the capture rate, and consequently of the total number N of WIMPs captured by the Sun, as a func-tion of the WIMP parameters is now easily understood. At fixed m, Nincreases linearly withp up to the saturation
level. For m⫽50 GeV, this behavior can be seen in Fig. 1
共coherent interactions兲 and Fig. 2 共spin-dependent
interac-tions兲, where the thick solid line represents Nas a function of p for the case of no-annihilation (a⫽0) and the thin
solid lines show N for some representative nonvanishing values of
具
av典
0. In calculating Nas a function ofp, wehave set ⫽l, i.e., ⫽1. Forl we have used, here and
throughout the paper, the default value l⫽0.3 GeV cm⫺3. The current uncertainty on l (0.2 GeV cm⫺3ⱗl ⱗ0.7 GeV cm⫺3, for an isothermal galactic halo 关7兴兲 may actually increase or reduce Nby at most a factor of 2. The no-annihilation line represents, for a given WIMP mass, the maximal number of WIMPs that can be captured by the Sun during its lifetime. If annihilation is present, Nis obviously smaller, depending on the strength of the annihilation cross section. Results very similar to those shown in Figs. 1 and 2 are obtained for different WIMP masses. Notice that the tran-sition from the linear regime to the saturation level occurs at quite different values for the relevant WIMP-nuclen cross section in the two limiting cases of pure coherent and pure spin-dependent interactions. This is again due to the fact that for WIMPs heavier than tens of GeV, scattering on heavy
nuclei in the Sun is enhanced for coherent interactions, and this shifts the transition from the linear to the saturation re-gime toward WIMP-nucleon cross sections smaller than in the spin-dependent case.
The previous discussion may be easily extended to the case in which the WIMP under consideration does not pro-vide the total amount of local dark matter共i.e.,⬍1). In this case, the relevant quantities are conveniently plotted in the plane p–N. For the case of coherent interactions the
re-sults are displayed in Fig. 3, whose four panels refer to dif-ferent representative values: ⫽1, 0.1, 0.01, 0.001. In each panel the dashed curve denotes a representative iso-␦ line with␦⫽1 and the solid line displays Nas a function ofp
for the limiting casea⫽0. For the sake of comparison, the top-left panel repeats the case already shown in Fig. 1 (
⫽1). The features of the panels with ⬍1 are easily
under-stood in terms of those represented in the top-left panel: 共i兲 the iso-␦ curve simply shifts horizontally to the left by an amount equal to , since ␦ depends on the scattering cross sectionpbut it does not depend on;共ii兲 the slanted part of
the solid line is unchanged since ⌫c in the linear regime, as given by Eq.共25兲, is a function of the productp; 共iii兲 the
flat part of the solid line共saturation regime兲 is lowered by an amount since⌫c,sat in Eq.共26兲 depends on but is
inde-pendent ofp. We explicitly consider the possibility of
sub-dominant WIMPs since this is a feature which can naturally occur for specific dark matter candidates, like, for instance, the neutralino, as it will be discussed in Sec. VII.
We are now able to understand which regions of WIMP parameter space potentially lead to observable modification of the solar structure, by comparing the iso-␦ lines with the curves which describe the number of WIMPs in the Sun. The first result is that the WIMP maximum effect occurs when annihilation is negligible and for scattering cross sections in the interval 10⫺38cm2ⱗcⱗ10⫺36cm2 for coherent inter-actions and 10⫺35cm2ⱗsdⱗ10⫺33cm2for spin-dependent interactions. These cross sections correspond to a situation in which the number of WIMPs in the Sun has essentially reached its saturation value and the WIMP energy transport is maximally efficient共i.e., K⬃K0). This is manifest in Figs. 1 and 2 for m⫽50 GeV and occurs also for all the WIMP mass range mⲏ30 GeV under study in the present paper. The value of ␦ which corresponds to this maximal effect is
␦⬃10, both for coherent and spin-dependent interactions.
We will use this information in the next sections where we will calculate the maximal deviation of the Sun properties when WIMPs are incorporated in its interior.
We stress that, for the WIMP mass range considered in the present paper, the process of WIMP capture cannot produce modifications of the Sun which refer to values of ␦ larger than about 10, for any value of the scattering cross section. Independent limits on the WIMP-nucleon interactions, like the ones that are obtained from direct search experiments, can further bound the values of␦ that can be reached. This is the case for coherent interactions, for which direct search experiments have already been able to set stringent bounds. For m⫽50 GeV, the upper limit oncis shown in Fig. 1
by a vertical dashed line. From this figure, we can therefore see that once the direct detection constraint is taken into
account, the largest values of␦which can be obtained are of the order of 10⫺6–10⫺7, which are far too small to lead to observable effects, as already discussed at the end of Sec. III. Once rescaling is considered, the behaviors of the iso-␦ and Nmax lines plotted in Fig. 3 show that ␦’s of the order of 10⫺3–10⫺2 can be reached. These values are close to the necessary 共but not sufficient兲 condition for WIMPs to have observable effects on the Sun 共i.e. ␦ⲏ10⫺2), introduced in Sec. III, and therefore they need to be analyzed in detail.
In the case of spin-dependent interactions, direct search experiments are also setting bounds onsd, but these limits
are not model independent since they rely on the way the interaction of WIMPs with the spin of the nucleus is realized
关5兴. For the representative case of a Z-boson mediated
inter-action and for m⫽50 GeV, the present upper limit is shown in Fig. 2 by a dotted vertical line. Since this line cannot be assumed to be a model independent bound, in the case of spin-dependent interactions we are left with the possibility to reach values of␦⬃10.
Up to this point we have quantified the amount of possible deviations induced by a generic WIMP in terms of the pa-rameter␦. We now turn to discuss the possibility to experi-mentally observe such modifications by means of helioseis-mology and of the boron solar neutrino flux.
V. WIMPS AND HELIOSEISMOLOGY
Helioseismology has provided very accurate information on the solar structure and it has been able to establish severe
constraints and tests of the solar standard model共SSM兲 cal-culations. For instance, helioseismology accurately deter-mines the depth of the convective envelope Rb and the photospheric helium abundance Yph. Moreover, by inversion
of helioseismic data, one can determine the共squared isother-mal兲 sound speed in the solar interior, u⫽P/, with high accuracy ( P and denote the pressure and density inside the Sun兲. In Fig. 4, we show the uncertainty in the helioseismic determination of u as a function of the radial coordinate R/R䉺. The light band corresponds to the so-called ‘‘3’’ errors which have been estimated conservatively by adding linearly all known individual uncertainties关18兴. If uncertain-ties are added in quadrature, the global error is about one third. This yields the so called ‘‘1’’ errors which are shown by the dark band in Fig. 4. This latter procedure was also used by Bahcall et al.关36兴 with similar results. For interme-diate values of the radial coordinate, the uncertainty in the squared sound speed is at the level of 10⫺3, while in the innermost part of the Sun, as well as on the surface, it is at the percent level. We remark that, as found in Refs.关18兴 and
关36兴, uncertainties corresponding to statistical errors of the
frequency measurements are generally much smaller than the ‘‘systematical uncertainties’’ of the inversion method. These latter include the choice of the reference model and of the free parameters in the inversion procedure. These uncertain-ties are particularly important close to the solar center, since there are very few p-modes that sample this region well. The errors quoted above include both statistical and systematic uncertainties. On the other hand, Refs.关37兴 and 关38兴 present
FIG. 3. The same as in Fig. 1, plotted as a function ofp, for
different values of the rescaling parameter . The solid lines de-note the total number of WIMPs accumulated in the Sun for a no-nannihilating WIMP. The dashed lines show the contours of iso-␦
for ␦⫽1. The vertical dashed
lines indicate the upper limit on
sound speed profiles with errors which look significantly smaller than our estimate. This occurs since ’’the quoted un-certainties include only the contribution arising from the fre-quency observations,’’ as clearly stated in关38兴.
Figure 4 also shows the difference between u as predicted by the SSM of Ref.关39兴 and the helioseismological determi-nation usun, normalized to usun. We notice that the SSM
accurately reproduces u for all the radial profile.
In order to understand whether helioseismology can tell us something about WIMPs, we have constructed solar mod-els for values of the WIMP parameters which maximize the role of WIMPs in the energetics of the Sun, irrespective of possible bounds derivable from direct search experiments. As discussed in the previous section, this corresponds to ␦
⬃10. For m⫽50 GeV, this is shown, for instance, by the large dot on the ‘‘no-annihilation line’’ in Fig. 1. We have taken into account the time evolution of the number of WIMPs in the Sun and we have described WIMP energy transport at each stage of Sun evolution according to Eq.
共12兲. The sound speed profiles obtained for the representative
WIMP masses m⫽30, 50 and 100 GeV are compared with the sound speed profile of the SSM in Fig. 5. Larger WIMP masses lead to even smaller effects. For any WIMP mass, smaller values of␦ clearly correspond to smaller effects.
The first observation is that, even in the most favorable situation, WIMPs produce a modification of the sound speed profile which is much smaller than the accuracy of present helioseismic determinations. The smallness of the effect is explained by two main reasons. First, WIMPs with masses larger than tens of GeV are confined in a small region of the Sun of radius Rⱗ0.02R䉺. In this region the relevant
physi-cal quantities 共temperature, luminosity, etc.兲 have small variations. As a consequence, even if WIMPs affect sizeably the gradient of these quantities 共e.g. the temperature gradi-ent兲, this translates into small modifications of their ‘‘global’’ radial profile. Second, a compensation occurs between two different effects. This can be easily understood in terms of the fact that, from the perfect gas law, which describes to a good approximation most of the solar core, one has u
⫽P/⯝T/, where is the mean molecular weight. WIMPs produce a quasi-isothermal core, thus decreasing the central temperature Tc of the Sun. At the same time, how-ever, with a lower Tcone has smaller nuclear reaction rates.
This translates into a slower helium production at the center of the Sun and, thus, into a decrease of the mean molecular weight. When considering u⯝T/, the effects of the varia-tions of T andcompensate, leaving the sound speed profile almost unaffected.
The results shown in Fig. 5 refer to a situation which maximizes the effect of the presence of WIMPs in the Sun:
␦⬃10. As discussed in the previous section, for
spin-dependent interactions, this value of␦ can be reached, while for coherent interaction ␦’s of the order of at most 10⫺3–10⫺2can be obtained. Our results show that in neither case 共spin-depenent nor coherent兲 can the presence of WIMPs in the Sun affect u to a level which is currently accessible by helioseismology.
We notice also that the necessary共but not sufficient兲 con-dition stated in Sec. III, i.e., that␦ must be larger than about
FIG. 4. Uncertainty bands in the reconstruction of the squared isothermal sound speed u in the Sun interior. The light region cor-responds to the conventional ‘‘3’’ error band, while the dark band identifies the conventional ‘‘1’’ range. The solid line corresponds to the fractional difference between u determined from observa-tional data and u calculated from the BP2000 solar model关39兴.
FIG. 5. Relative difference of the squared isothermal sound speed in the Sun interior u calculated for a standard solar model and for solar models with accreting WIMPs, as a function of the radial coordinate R/R䉺. The light region corresponds to the conventional ‘‘3’’ error band, while the dark band identifies the conventional ‘‘1’’ range. The dotted, dot-dashed and dashed lines refer to solar models with accreting WIMPs of masses m⫽30, 50, 100 GeV and coherent scattering cross sections which, for each mass, maximize the effect of the presence of WIMPs in the Sun.
10⫺2 in order to possibly have an observable effect, may be reinforced: in fact our detailed calculation shows that, for WIMP masses above 30 GeV, not even␦⬃10 is sufficient to produce a currently observable effect.
In summary, the results of this section show that no sig-nificant information about WIMPs with masses m
ⲏ30 GeV can be obtained at present from helioseismic data,
for any value of the WIMP-nucleon cross section.
Conclusions conflicting with ours were derived in Ref.
关29兴. This may be traced back to the fact that in Ref. 关29兴 the
accuracy of the squared sound speed is taken to be of about 0.1% also in the innermost part of the Sun core relevant for WIMP confinement, say Rⱗ0.02R䉺. We argued above that, at these small radii, the accuracy of the helioseismic deter-mination is significantly worse.
VI. WIMPS AND THE BORON NEUTRINO FLUX From the previous discussion, one could expect that the Sun central temperature Tcis more sensitive than the central
sound speed to the presence of WIMPs in the Sun. This is actually the case, as can be seen in Table II, which shows that WIMPs produce variations of Tc at the percent level,
while the central sound speed uc is affected at most at the
level of 10⫺3.
Unfortunately, the central temperature of the Sun is not directly observable. The most direct information on it comes from the measurements of the solar neutrino fluxes. In par-ticular, the comparison between the SNO charged current and super-Kamiokande data 关21–26兴 has allowed to deter-mine, in a model independent way, the 8B neutrino flux pro-duced in the Sun with an accuracy of about 20%.2 This re-sult, under the assumption that the 8B neutrino flux scales with the Sun central temperature as⌽B⬀Tc
20
, can be used to determine Tcat the 1% level关28兴.
However, the presence of WIMPs in the core of the Sun
produces a very peculiar modification of the temperature profile for which the scaling law ⌽B⬀Tc
20
is not valid. WIMPs can sizeably affect the temperature profile only in the innermost part of the Sun, and in particular in a region which is smaller than the boron neutrino production region (rB⯝0.05R䉺). Figure 6 shows the radial profile of the Sun temperature in the SSM and in models of the Sun where WIMPs with masses of 30, 50 and 100 GeV are present. Again, we have chosen the WIMP parameters which maxi-mize the effect of their presence in the Sun. Figure 6 clearly shows that even though the central temperature of the Sun can be lowered by the presence of WIMPs at the percent level 共as it is reported also in Table II兲, nevertheless the temperature at rB⯝0.05R䉺, where the boron neutrino pro-duction is maximal, is left practically unchanged.
In order to perform a quantitative analysis, one has there-fore to look directly at the boron neutrino flux and not simply at the central temperature Tc. If this is done, it turns out that the solar models of Fig. 6, which correspond to WIMP masses equal to 30, 50 and 100 GeV, predict variations of the boron neutrino flux equal to␦⌽B/⌽B⯝⫺3.6%, ⫺1.0% and
⫺0.1%, respectively. These variations are well below the
accuracy of the present determination of the boron neutrino flux. We remind the reader that, for each WIMP mass, we have optimized the other WIMP parameters in order to maxi-mize the effect. Larger masses and WIMP annihilation would lead to even smaller deviations.
In conclusion, our results allow us to conclude that no information about WIMPs with massess mⲏ30 GeV can be obtained from our present knowledge of the boron neutrino flux.
A previous independent analysis 关30兴 concluded that WIMPs lighter than 60 GeV are in conflict with the present accuracy in the determination of the boron neutrino flux, based on the fact that the presence of WIMPs in the core of the Sun can modify the central temperature Tc at the 1%
level. We should stress that this lower limit on the WIMP 2The SNO experiment has recently reported the observations of
neutral current interactions on deuterium 关27兴 confirming the 8B neutrino flux predicted by SSM with an accuracy of about 12%. This result is obtained under the assumption that the solar neutrino energy spectrum is undistorted. For this reason, in our analysis, we refer to the result obtained by comparing SNO charged current and super-Kamiokande data, which instead do not rely on this assump-tion, see Refs.关21,40兴.
TABLE II. Fractional variations of the central temperature of the Sun Tc, of the squared isothermal sound speed in the Sun interior and of the 8B neutrino flux, calculated in solar models where WIMPs are accreted in the Solar core. For each model, the WIMP-nucleon scalar cross section has been set at the value which maximizes the effect of the presence of WIMPs in the Sun.
m 共GeV兲 ␦Tc/Tc ␦uc/uc ␦⌽B/⌽B
30 ⫺2.1⫻10⫺2 ⫺4⫻10⫺3 ⫺4.5⫻10⫺2
50 ⫺1.2⫻10⫺2 ⫺2⫻10⫺3 ⫺1.3⫻10⫺2
100 ⫺5.2⫻10⫺3 ⫺6⫻10⫺4 ⫺2.1⫻10⫺3
FIG. 6. Temperature profile of the standard solar model and of models of the Sun where WIMPs with masses of 30, 50 and 100 GeV are present. The coherent WIMP-nucleus scattering cross sec-tions are chosen, for each mass, in order to maximize the effect of the presence of WIMPs in the Sun.
mass does not apply, since, as we have discussed above, WIMPs, though capable of changing Tcat the percent level,
nevertheless are too concentrated in the interior of the Sun to produce sizeable modifications of the boron neutrino flux, which is the quantity experimentally measured.
VII. ONE REALIZATION OF WIMP: THE NEUTRALINO It is instructive to discuss how some of the previous prop-erties are realized in the case of one of the favorite WIMP candidates: the neutralino. In Figs. 7–9 we show some of the relevant properties.
The supersymmetric scheme we are adopting here is an effective minimal supersymmetric extension of the standard model 共MSSM兲. Its parameters are defined directly at the electroweak scale and represent the minimum set necessary to shape the essentials of the theoretical structure of the model and of its particle content. We refer to Ref. 关41兴 for details on the theoretical aspects and on the updated experi-mental bounds.
The neutralino-nucleon cross sections, both coherent and spin-dependent, are calculated in terms of the couplings and supersymmetric particle masses as discussed, for instance, in Refs. 关5,10兴. The parameter is evaluated by using a stan-dard rescaling procedure 关42兴 which relates the fraction of local neutralino dark matter in the Galaxy to its fractional abundance in the Universe, i.e., to its relic abundance⍀h2:
⫽min
冋
1, ⍀h 2共⍀mh2兲min
册
, 共27兲
where (⍀mh2)minis the minimum value of⍀mh2compatible
with halo properties. Here we set (⍀mh2)
min⫽0.05. The present-day neutralino relic abundance has been calculated according to Ref.关43兴.
To establish some relevant order of magnitude for the annihilation cross sectionait is useful to recall the
approxi-mate expression⍀h2⬃3⫻10⫺27cm3 s⫺1/
具
av典
int, where具
av典
int denotes the thermal average of the product(a•v) integrated from the freeze-out temperature Tf to the present-day one. For instance the upper bound ⍀h2ⱗ0.3 translates into a lower bound
具
av典
intⲏ10⫺26cm3 sec⫺1.In Fig. 7 we display the correlation between and
具
av典
0, calculated by varying the MSSM parameters and applying to the model the relevant experimental constraints. Figure 7 shows that for a neutralino in the MSSM, the res-caling factor is always smaller than 1 for values of具
av典
0 larger than about 10⫺27–10⫺26cm3 s⫺1. Notice that具
av典
0 denotes the thermal average of the product (a•v) in the Sun’s core, and does not coincide with the relevant thermal average which enters in the calculation of the relic abun-dance.We turn now to the graphic representations in the plane
p-N. Figure 8 共Fig. 9兲 shows the scatter plot of the
MSSM configurations for which the total capture rate is
FIG. 7. Rescaling parameter⫽min(1,⍀h2/0.05) as a function of the zero-temperature thermally averaged neutralino self-annihilation cross section 具av典0 in the effective minimal
super-symmetric standard model共MSSM兲.
FIG. 8. Total number Nof neutralinos accumulated in the Sun as a function of c in the effective MSSM. Only configurations where the total capture rate is dominated by coherent interactions are displayed. Neutralino masses are in the range 50 GeV⬍m
⬍500 GeV. Dots denote configurations where the neutralino is the
dominant dark matter component: 0.05⬍⍀h2⬍0.3 共i.e., ⫽1).
Crosses refer to configurations where the neutralino is a subdomi-nant dark matter component:⍀h2⬍0.05 共i.e.,⬍1). The slanted
solid line shows N for m⫽50 GeV and for the limiting case of nonannihilating WIMPs. The dot-dashed lines denote iso-␦ con-tours. For each pair, the upper line refers to the iso-␦ curve for
m⫽50 GeV and⫽1, while the lower line refers to the same
iso-␦ and, for m⫽500 GeV. The vertical dashed lines indicate the upper limit onpfor coherent interactions.
dominated by coherent共spin-dependent兲 interactions and for the neutralino mass range: 50 GeV⬍m⬍500 GeV, the lower bound arising from experimental constraints on the MSSM. The points are labeled according to the values of the neutralino relic abundance: dots denote configurations where the neutralino is the dominant dark matter component: 0.05
⬍⍀h2⬍0.3, i.e., ⫽1; crosses refer to configurations where the neutralino is a sub-dominant dark matter compo-nent: ⍀h2⬍0.05, i.e., ⬍1. The slanted solid line shows Nfor m⫽50 GeV and for the limiting case of nonannihi-lating WIMPs. This line represents, for each of the two lim-iting cases of coherent and spin dependent domination, the actual upper limit for N for neutralinos in the effective MSSM, due to the lower bound on the neutralino mass quoted above. The dot-dashed lines denote iso-␦ contours for two representative values of␦. For each pair, the upper line refers to m⫽50 GeV and⫽1, while the lower line refers to m⫽500 GeV and⫽1.
By comparing the scatter plot with the iso-␦ lines, we can notice that the configurations with no rescaling (⫽1) are compatible with values of ␦ smaller than about 10⫺9, both for coherent and spin-dependent interactions. We remind that these are configurations with large neutralino relic abun-dance. In the case of configurations with ⬍1, we have to consider the shift of the iso-␦ lines shown in Fig. 3 and the range of values for in the MSSM shown in Fig. 7. In this case, we obtain that the maximal effect induced by a sub-dominant relic neutralino is for values of ␦ of the order of 10⫺6 for the coherent case and 10⫺4for the spin-dependent one.
We can therefore conclude that the maximal effects in-duced in the Sun by relic neutralinos are much smaller than those valid for a generic WIMP and discussed in Sec. IV.
This implies that the specific case of the neutralino has even fewer chances to be bounded by the study of solar properties.
VIII. CONCLUSIONS
Our understanding of solar physics has significantly ad-vanced in recent times, because of remarkable improvements in helioseismology and of new data on solar neutrinos. In the present paper we have addressed the question of whether these new developments put solar physics in a position to provide constraints on the possible presence of WIMPs in the core of the Sun, for values of WIMP parameters under cur-rent exploration in WIMP searches. We summarize here our main results:
We have provided a quantitative criterium to determine whether a putative WIMP candidate could produce observ-able modifications of the solar structure. Namely, we have introduced the parameter ␦, defined as the ratio between the WIMP luminosity and the radiative luminosity at the center of the Sun 关see Eq. 共17兲兴, which can be analytically calcu-lated as a function of the WIMP-nucleon scattering cross section p and the number of WIMPs in the Sun N. By
considering the uncertainty in the radiative opacity, one finds that␦ⲏ10⫺2is a necessary共but not sufficient兲 condition for WIMPs to have observable effects on the Sun.
We have calculated the number of WIMPs in the Sun for a generic WIMP candidate as a function of the WIMP-nucleon scattering cross section p and of the WIMP-antiWIMP pair annihilation cross sectiona, both for
coher-ent and spin-dependcoher-ent WIMP interactions. We have shown that a value ␦⬃10 can be reached, if WIMP-antiWIMP an-nihilation is negligible andp⬃10⫺37cm2共coherent scatter-ing兲 orp⬃10⫺34cm2共spin dependent interactions兲. For co-herent WIMP scattering, such large values for the cross sections are excluded by direct search experiments, unless WIMPs give a subdominant contribution to dark matter of our galaxy. In this case, if one takes into account direct search bounds (pⱗ10⫺41cm2, where is the rescaling parameter which accounts for the actual fraction of local dark matter to be ascribed to the given WIMP兲, one obtains
␦ⱗ10⫺6 in the assumption of no rescaling (⫽1) and ␦
ⱗ10⫺3 in the general case (⭐1).
We have considered the neutralino as a specific WIMP candidate. We have calculated the number of neutralinos in the Sun for MSSM configurations compatible with present experimental and cosmological constraints. For each con-figuration, we determined the rescaling parameter by the standard rescaling procedure described in Eq. 共27兲. In the case of no rescaling (⫽1), we obtained␦ⱗ10⫺9. For sub-dominant neutralinos (⭐1), we obtained␦ⱗ10⫺6for con-figurations in which the capture rate is dominated by coher-ent scattering and ␦ⱗ10⫺4 for configurations in which the dominant contribution is due to spin dependent interactions. The previous points already show that solar physics is not competitive with direct experiments in a large part of the WIMP parameter space under current exploration in WIMP searches. In order to complete our analysis and to understand whether some information could be obtained from the present helioseismic and solar neutrino data, we have
con-FIG. 9. The same as in Fig. 8, for MSSM configurations where the total capture rate is dominated by spin-dependent interactions.
structed solar models, for WIMP masses above 30 GeV, choosing the value of the WIMP parameters which maximize the effect of WIMPs on the Sun (␦⬃10). As a result of the presence of WIMPs in the Sun, we obtained variations of the sound speed profile and of the 8B neutrino flux which are within the current experimental uncertainties. The smallness of the effects is essentially due to the the smallness of the WIMP extension region which, for WIMPs with masses larger than tens of GeV, is r⭐0.02R䉺.
In conclusion, no constraints can be derived at present from solar physics for WIMPs with masses above 30 GeV. Our conclusions are at variance with results derived in Refs.
关29,30兴. The origins of these disagreements have been
eluci-dated in the present paper.
APPENDIX
We give here for completeness the expression of the fac-tor Si, introduced in Eq.共25兲, as derived from Ref. 关35兴. It is
given by
Si⫽
具
Fi典
i具
典
i 共⬁兲, 共A1兲
where the brackets indicate an average over the mass density profile i(r) of the ith nucleus in the Sun:
具
f典
i⬅ 1 Mi冕
0R䉺
4r2dri共r兲f 共r兲. 共A2兲 All the quantities are defined in Sec. III with the exception of Fi, which is given by Fi⫽ v ¯2 vesc 2 1 3b
再
关共⫺ˆ ,ˆ兲⫺共Aˆ⫺,Aˆ⫹兲兴 exp共⫺aˆ2兲 共1⫹a兲1/2 ⫺关共⫺ˇ ,ˇ兲⫺共Aˇ ⫺,Aˇ⫹兲兴 exp共⫺bˇ2兲 共1⫹b兲1/2 ⫻exp关⫺共a⫺b兲A2兴冎
. 共A3兲The definition of the quantities a, b,, A, Aˆ⫹,⫺, Aˇ⫹,⫺,ˆ , andˇ may be found in Ref.关35兴.
关1兴 See, for instance, Y. Sofue and V. Rubin, astro-ph/0010594;
S.L. Bridle et al., astro-ph/9903472.
关2兴 See, for instance, S. Esposito, G. Mangano, G. Miele, and O.
Pisanti, Nucl. Phys. B568, 421共2000兲; E. Lisi, S. Sarkar, and F.L. Villante, Phys. Rev. D 59, 123520共1999兲; G. Fiorentini, E. Lisi, S. Sarkar, and F.L. Villante, ibid. 58, 063506共1998兲.
关3兴 See, for instance, N. Fornengo, Nucl. Phys. B 共Proc. Suppl.兲 110, 26共2002兲; hep-ph/0201156; A. Bottino and N. Fornengo,
hep-ph/9904469.
关4兴 For a review of WIMP direct searches see, for instance, A.
Morales, Nucl. Phys. B共Proc. Suppl.兲 110, 39 共2002兲.
关5兴 A. Bottino, F. Donato, G. Mignola, S. Scopel, P. Belli, and A.
Incicchitti, Phys. Lett. B 402, 113共1997兲.
关6兴 P. Belli, R. Bernabei, A. Bottino, F. Donato, N. Fornengo, D.
Prosperi, and S. Scopel, Phys. Rev. D 61, 023512共2000兲.
关7兴 P. Belli, R. Cerulli, N. Fornengo, and S. Scopel, Phys. Rev. D 66, 043503共2002兲.
关8兴 DAMA/NaI Collaboration, R. Bernabei et al., Phys. Lett. B 424, 195共1998兲; 450, 448 共1999兲; 480, 23 共2000兲.
关9兴 A.K. Drukier, K. Freese, and D.N. Spergel, Phys. Rev. D 33,
3495共1986兲; K. Freese, J.A. Frieman, and A. Gould, ibid. 37, 3388共1988兲.
关10兴 A. Bottino, F. Donato, N. Fornengo, and S. Scopel, Phys. Lett.
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